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Question Number 55230 by maxmathsup by imad last updated on 19/Feb/19

$$calculate{lim}_{\xi \to 0}{\int }_1^{1+\xi }\frac{arctan\left(\xi t\right)}{t}dt.$$

Commented by maxmathsup by imad last updated on 25/Feb/19

$$\mathrm{\exists }\alpha \in ]1,1+\xi [/A\left(\xi \right)={\int }_1^{1+\xi }\frac{arctan\left(\xi t\right)}{t}dt=arctan\left(\xi \alpha \right){\int }_1^{1+\xi }\frac{dt}{t}$$$$=arctan\left(\xi \alpha \right)ln\mid 1+\xi \mid \Rightarrow {lim}_{\xi \to 0}A\left(\xi \right)=arctan\left(0\right)ln\left(1\right)=0\Rightarrow$$$${lim}_{\xi \to 0}{\int }_1^{1+\xi }\frac{arctan\left(\xi t\right)}{t}dt=0.$$

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